For instance, when the first homology group [1]} \(H_1(M)\)  of the 3-manifold \(M\)  is trivial, one can compute [2]} the observables of the abelian Chern-Simons theory defined in \(M\)  by means of perturbation theory. The action for the connection \(A\)  is given by \(2 \pi k \int A \wedge dA \)  and the variation of the action with respect to the fields is proportional to the curvature \(F_A= dA\) .
By introducing the coupling \(\int B \wedge dA \)  of the curvature with an external classical source \(B = B_\mu (x) dx^\mu \) , one finds
\(\widetilde{Z}_{SD} [B] \equiv \left\langle e^{ 2 \pi i \int B \wedge dA } \right\rangle = e^{ -i \, \pi / 2 k \int B\wedge B} \; ,\) 
